Download Polynomials and Equations and Inequalities

Regents Review Session #1 Polynomials
Adding and Subtracting Polynomials
What do you do?
1. Add/subtract like terms
Example:
1. (8x3 - 9x2 + 6x + 2) - (-7x3 - 5x2 + 1x - 8)
Multiplying With Polynomials
What do you do?
1. Distribute (or double-distribute/FOIL, when necessary)
2. Combine like terms
Examples:
1. 4x2 - 6x(2x - 7) + 5 - 8x(x + 2)
3. (7x - 2)2
2. (4y - 3)(5y2 + 3y - 10)
**** 4.
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Factoring
What do you do?
1. Always look for a greatest common factor first.
2. Factor by into two binomials - be sure those binomials are also in simplest factored
form.
(Remember - if there is no greatest common factor in the given trinomial, there cannot
be a common factor in the parentheses)
­
Be aware of the difference of perfect squares, sum/difference of two cubes, and
factoring by grouping.
Examples:
1. x4 - 16
2. 6x3 + 14x2 - 9x - 21
3. 4x2 - 5x - 6
4. 75x3 - 200x2 - 3x + 8
2
Simplifying Algebraic Fractions
What do you do?
1. Factor the numerator, factor the denominator, and cancel common factors
between the numerator and denominator.
Examples:
b.
6x2 - 5x - 4
10-2x
a.
9x2 - 16
x2 - 2x - 15
c.
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Multiplying and Dividing Algebraic Fractions
What do you do?
1. Factor EVERYTHING
2. Cancel any common factors that appear in a numerator and a denominator.
REMEMBER: You are only allowed to cancel factors, factors are connected by a
multiplication sign. YOU CANNOT CANCEL ACROSS AN ADDITION OR
SUBTRACTION SIGN.
3. Multiply numerators, leave in factored form, and then multiply denominators.
** IF you are dividing, you must turn the division into multiplication and take the
reciprocal of the 2nd fraction.
Examples:
4
2.
Express in simplest form:
3. Multiply x3 + 4x2 - 9x - 36 *
x2 - 2x - 15
2x - 10
3x - x2
5
Adding and Subtracting Algebraic Fractions
What do you do?
Factor all denominators.
1.
2. Find the LCD between all of the denominators.
3. Make the LCD the denominator on all fractions and
appropriately adjust the numerators.
4. Keep the common denominator and add/subtract the
numerators.
5. Simplify the expression, if possible.
Examples:
+
x
1.
x-3
2
x
2.
5x + 7
x2 - 4
-
3
4x + 8
6
3.
9
+
2
x + 6x + 8
7
x + 4x + 4
2
4.
x+1
x + 2x - 15
2
+
x - 4
x -x-6
2
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Complex Fractions
What do you do?
There are two ways to do these problems:
Method 1:
1. Get a common denominator within the numerator, adjust the numerators, and
write the fractions as one fraction.
2. Get a common denominator within the denominator, adjust the numerators, and
write the fractions as one fraction.
3. Keep the top fraction, change the operation from division to multiplication, flip
the bottom fraction.
4. Factor and simplify, if possible.
Method 2:
1. Find the LCD for all denominators within the complex fraction.
2. Multiply each term by the LCD (if this is done correctly, you will eliminate the
fractions within the fraction.
3. Factor and simplify, if possible.
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Examples:
1.
2.
3. 9
Equations and Inequalities
I. Undefined Fractions
When is the fraction undefined? State the domain using interval notation.
1. Be sure the fraction is in simplest form and then set the denominator equal to zero and solve. Those solutions must be restricted (removed) from the domain. Remember for interval notation: Brackets mean included, parenthesis mean not include, positive and negative infinity can never be 'included'
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II. Factoring ­ Discussed Earlier ­ To solve using factoring: be sure one side of the equations is equal to zero, factor and then set each factor equal to zero and solve. 11
III. Complete the Square
Process for Completing the Square
1. Be sure the quadratic coefficient is 1.
2. Move constant to the other side of the equation.
3. Take 1/2 of the linear coefficient, square it, add it to both sides of the equation.
4. Rewrite the one side of the equation as a quantity squared, i.e. (x ± ?)2
5. Take the square root of both sides ­ (remember, there is a positive and a negative square root of a number).
6. Solve for x. Be sure the radical is in simplest radical form. 1. x2 + 8x ­ 3 = 0
2. 2x2 ­ 5x + 1 = 0
3. 2x2 ­ 7x ­ 14 = 0
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Quadratic Formula
Must be sure to have one side equal to zero, before you solve the equation. Do not approximate the radical unless you are told to do so. 13
Solving Algebraic Fractions
1. Either multiply every term by the LCD (to clear the fractions) or get a common denominator for every term, so you can then focus on the numerators only (since the denominators are all equal now, you may 'drop' the denominators).
2. Solve the resulting equation.
3. Check your solutions for a possible undefined fraction. 14
Absolute Value Equations
How to solve absolute value equations:
1. Get the absolute value term alone.
2. Set what is inside the absolute value sign equal to the other side of the equation. Then, set what is inside the absolute value sign equal to the negation of the other side of the equation. 3. Solve each equation for x.
4. Check the values for x to make sure they work.
1. |2x + 5| = 4x + 3
2.
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Inequalities
Simple ­Linear
Remember: when you multiply or divide by a negative, you must switch the direction of the inequality sign.
Solve and write the solution using interval notation.
1. ­3 < 2x + 5 < 21
2. ­4x ­ 9 > 15 OR 8 ­ 3x < 11
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Quadratic Inequalities
How do you solve a quadratic inequality?
1. Solve the corresponding equation ­ those solutions are the boundary points. The boundary points split the number line into intervals.
2. Plot the solutions you found in step one on a number line.
3. Pick a number from each interval that was created when you completed step 2.
4. Test each of the values you chose in step 3 in the original inequality.
5. If the test gives a true statement, that interval contains solutions to the inequality. Shade the intervals on the number line that result in a true statement.
6. Write your solution set using interval notation and/or graph the solution set.
1. 0 < 6x2 ­ 7x ­ 20
2. The area of a rectangular garden must not exceed 180 square feet. The length of the rectangular garden is 3 more than twice the width. What are the largest possible integer dimensions of the garden?
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Absolute Value Inequalities
How do we solve absolute value inequalities?
1. Be sure the absolute value is alone.
2. You will need to solve two inequalities ­ one will be exactly what you see with the absolute value signs removed and the other will be exactly what is in the absolute value signs with the inequality sign 'flipped' and the number negated.
3. Solve each and state your solution as requested.
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Rational Inequalities
To solve rational inequalities
1. Write the inequality as an equation and solve.
2. Determine any/all places where the fraction is undefined (Be sure the fraction is in simplest form).
3. On a number line, mark each of the values determined from steps 1 and 2. These are the boundary points that create the possible intervals of solution.
4. Select a point from each of the intervals, created by the boundary points, and check to find the interval(s) that satisfy the given inequality.
5. Shade appropriately.
6. Write your solution using interval notation. 19
Solve, graph and write the solution using interval notation.
x3 ­ 8x2 ­ 4x + 32 < 0
x + 5
20